A homological approach to pseudoisotopy theory. I

Abstract

We construct a zig–zag from the once delooped space of pseudoisotopies of a closed 2n-disc to the once looped algebraic K-theory space of the integers and show that the maps involved are p-locally (2n-4)$$(2n-4)$$-connected for n>3$$n,{>},3$$ and large primes p. The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik–Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa’s stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification is a calculation of the rational homotopy groups of BDiff∂(D2n+1)$$\mathrm {BDiff}_\partial (D^{2n+1})$$ in degrees up to 2n-5$$2n-5$$.